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This paper develops matrix-multiplication-based iterative refinement for diagonalizable non-Hermitian eigendecompositions. The main theory concerns simple eigenvalues and distinguishes two input regimes. In the right-only regime, where only…
The power method and block Lanczos method are popular numerical algorithms for computing the truncated singular value decomposition (SVD) and eigenvalue decomposition problems. Especially in the literature of randomized numerical linear…
To conduct a more in-depth investigation of randomized solvers for solving linear systems, we adopt a unified randomized batch-sampling Kaczmarz framework with per-iteration costs as low as cyclic block methods, and develop a general…
For compact self-adjoint operators in Hilbert spaces, two algorithms are proposed to provide fully computable a posteriori error estimate for eigenfunction approximation. Both algorithms apply well to the case of tight clusters and multiple…
Many optimization problems require hyperparameters, i.e., parameters that must be pre-specified in advance, such as regularization parameters and parametric regularizers in variational regularization methods for inverse problems, and…
Due to the ever growing amounts of data leveraged for machine learning and scientific computing, it is increasingly important to develop algorithms that sample only a small portion of the data at a time. In the case of linear least-squares,…
We consider a singular perturbed eigenvalue problem for Laplace operator in a cylinder with frequent interchange of type of boundary condition on a lateral surface. These boundary conditions are prescribed by partition of lateral surface in…
In this paper, we compute the structured eigenvalue backward error of a Rosenbrock system matrix $S(z)=\left[\begin{array}{cc} A-zI & B \\ C & P(z) \end{array}\right]$ for a given scalar $\lambda\in \mathbb C$. We have developed simplified…
In this paper we present a convergence rate analysis of inexact variants of several randomized iterative methods. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic…
The theoretical analysis of spectral clustering mainly focuses on consistency, while there is relatively little research on its generalization performance. In this paper, we study the excess risk bounds of the popular spectral clustering…
For a given subspace, the Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the Rayleigh-Ritz…
This paper proposes hybrid high-order eigensolvers for the computation of guaranteed lower eigenvalue bounds. These bounds display higher order convergence rates and are accessible to adaptive mesh-refining algorithms. The involved…
We consider filtered subspace iteration for approximating a cluster of eigenvalues (and its associated eigenspace) of a (possibly unbounded) selfadjoint operator in a Hilbert space. The algorithm is motivated by a quadrature approximation…
We derive new perturbation bounds for eigenvalues of Hermitian matrices with block structures. The structures we consider range from a standard 2-by-2 block form to block tridiagonal and tridigaonal forms. The main idea is the observation…
This work introduces the first small-loss and gradual-variation regret bounds for online portfolio selection, marking the first instances of data-dependent bounds for online convex optimization with non-Lipschitz, non-smooth losses. The…
A deflated and restarted Lanczos algorithm to solve hermitian linear systems, and at the same time compute eigenvalues and eigenvectors for application to multiple right-hand sides, is described. For the first right-hand side, eigenvectors…
We obtain Riesz transform bounds and characterise operator-adapted Hardy spaces to solve boundary value problems for singular Schr\"odinger equations $-\mathrm{div}(A\nabla u)+aVu=0$ in the upper half-space $\mathbb{R}^{1+n}_{+}$ with…
The standard approach for finding eigenvalues and eigenvectors of matrix polynomials starts by embedding the coefficients of the polynomial into a matrix pencil, known as linearization. Building on the pioneering work of Nakatsukasa and…
We present a variety of projection-based linear regression algorithms with a focus on modern machine-learning models and their algorithmic performance. We study the role of the relaxation parameter in generalized Kaczmarz algorithms and…
We derive bounds on the eigenvalues of a generic form of double saddle-point matrices. The bounds are expressed in terms of extremal eigenvalues and singular values of the associated block matrices. Inertia and algebraic multiplicity of…